Some examples of quasiisometries of nilpotent Lie groups

TL;DR

This paper constructs new quasiisometries of nilpotent Lie groups, including examples that are not close to automorphisms and biLipschitz maps of the Heisenberg group with non-vertical images.

Contribution

It introduces novel quasiisometries of nilpotent Lie groups that differ from automorphisms and constructs biLipschitz maps with unique geometric properties.

Findings

Existence of quasiisometries not close to automorphisms

Construction of biLipschitz maps sending vertical lines to non-vertical curves

New examples of geometric transformations in nilpotent Lie groups

Abstract

We construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group N, we construct quasiisometries from N to itself that is not at finite distance from any map that is a composition of left translations and automorphisms. We also construct biLipschitz maps of the Heisenberg groups that send vertical lines to non-vertical curves.